The present invention relates to the design of optical filters for use in spectroscopy analysis systems.
As described in U.S. Pat. No. 6,198,531, the entire disclosure of which is incorporated by reference herein, methods are known by which to construct optical filters having predetermined transmission spectra. In general, such an optical filter is an interference filter having alternating layers of high and low refractive index materials. By properly selecting the materials and layer thicknesses, the filter can be made to selectively pass predetermined percentages of incident light at different wavelengths, where the transmission percentage varies over the wavelength range.
One method of designing such a filter begins with a desired transmission spectrum and a selection of materials for the filter layers. Given this information, an algorithm optimizes the number of layers and their thicknesses to define a filter having a transmission spectrum that differs from the desired spectrum by an amount within an acceptable error tolerance. The procedure relies upon a merit function (“MF”) that describes the difference between the desired spectrum and any other spectrum:MF=ΣW(Tc−Tt)k/W,  (equation 1a)where Tc is the transmission percentage of the “other” transmission spectrum at a given wavelength, Tt, is the transmission percentage of the desired, or “target,” spectrum at the same wavelength, W is the number of wavelength channels measured and used in the merit function, and k is a constant. As should be understood in the art, constant k is a factor that determines the weighting applied in the MF to higher order differences between Tc and Tt. If k is equal to 1, the merit function calculates average error between the two transmission spectra. A constant of 2 provides a mean square error.
Other merit functions may be used, for example:MF=(1/W)ΣW((Tc−Tt)k/TOl)1/k,  (equation 1b)where TOl is the design tolerance. In another merit function,MF(x)=Σm(Ro(m)−R(x,m))2,  (equation 1c)where x is the vector of design variables, Ro is the specified reflectance at wavelength m, and R is the computed value of reflectance at m for particular values of x.
To determine the “other” transmission spectrum, the user makes an initial guess of the number of filter layers and their thicknesses. The layer materials, the substrate material and thickness, and the incident material (i.e. the material through which the light sample passes in travelling to the filter—typically air) are selected as initial conditions. The transmission spectrum of the resulting optical filter, constructed according to the design defined by the initial conditions and the initial guess, is determined. Procedures for determining a transmission spectrum from such information are known and are described in more detail below.
Referring to FIG. 1, if the thickness of each layer defines an axis in a coordinate system, a plot of the merit function as a function of layer thickness produces a cup-shaped surface 8 that is parabolic in any given cross-section parallel to the merit function axis. Thus, if the thicknesses of layers 1 and 2 are O and P, respectively, the error between the resulting transmission spectrum and the desired spectrum, as defined by the merit function, is the MF axis value of point B on surface 8. Once on surface 8, a quasi-Newton method of nonlinear optimization may be used to find points on surface 8 that are successively closer to the surface's minimum point A, eventually finding point A or a point acceptably close to point A.
Point A corresponds to thicknesses M and N of layers 1 and 2, respectively. The merit function value at these thicknesses, using the assumed materials and substrate, is not zero. That is, it is impossible under the assumed conditions to achieve the desired transmission spectrum. If the error represented by this merit function value is unacceptable, a second guess is made that includes a third layer of a material with a refractive index different from that of the second layer. The third guess may begin with the thicknesses determined in the first iteration for the first two layers, or the user may choose three new thicknesses. In either event, the algorithm re-optimizes for all three layer thicknesses, and the third guess therefore results in a four-dimensional space. The algorithm proceeds in the same manner outlined above, except with an additional thickness variable, and continues to add layers until the merit function value falls within the acceptable tolerance, thereby establishing the final filter design. A computer program distributed under the name TFCALC by Software Spectra, Inc. of Portland, Oreg. may be used to execute this iterative method of designing thin film layers.
A procedure for determining a transmission spectrum of an optical filter defined by an assumption of layer thicknesses and selection of layer materials, substrate material, substrate thickness and incident medium is based on a matrix method. Each filter layer is associated with a matrix Y1 as follows:
                              Y          l                =                                                            cos                ⁢                                                                  ⁢                δ                                                                    i                ⁢                                  /                                ⁢                n                ⁢                                                                  ⁢                sin                ⁢                                                                  ⁢                δ                                                                                        i                ⁢                                                                  ⁢                n                ⁢                                                                  ⁢                sin                ⁢                                                                  ⁢                δ                                                                    cos                ⁢                                                                  ⁢                δ                                                                        (                  equation          ⁢                                          ⁢          2                )            n is a complex refractive index:n=n1+ik1,  (equation 3)where n1 is the real refractive index, and k1 is the absorption coefficient. δ is a phase factor defined as:δ=(2Πn1k1x/λ)cos θ1,  (equation 4)where λ is wavelength, x is layer thickness, and θ1 is the interior angle of propagation in layer 1. For normally incident light, cos θ1=1. For tilted filters, however, in which the incident angle is oblique,cos θ1=(1−(no2 sin2θo)/n12)0.5,  (equation 5)where n0 is the refractive index of the incident medium (e.g. air), and θ0 is the angle of incidence in radians. The characteristic matrix of the entire film stack (M) is determined by multiplying the individual layer matrices:M=Y1×Y2× . . . ×Yl,  (equation 6)where layer 1 is the layer closest to the substrate.
Light transmitted through the film layers includes an electric vector component, E, and a magnetic vector component, H. The characteristic matrix M relates the electric and magnetic fields at individual layer boundaries as follows:
                                                                                  E                                                                    H                                                              =                  M          ⁢                                          ⁢          x          ⁢                                                                                  1                                                                                                                        n                      s                                        +                                          i                      ⁢                                                                                          ⁢                                              k                        s                                                                                                                                                                (                  equation          ⁢                                          ⁢          7                )            where ns and ks are the substrate's refractive index and absorption coefficient.
The propagation of light through a single thin film coating is illustrated in FIG. 11. As indicated in the figure, there are multiple reflections of incident light at the air/film and substrate/air interfaces. The transmittance and reflectance at the air/film interface are:T1=(4nsn)/(n+ns)2, and  (equation 8)R1=1−T1.  (equation 9)For normally incident light, n=no=1, where n is the refractive index. When the filter is tilted, however, the propagated wave splits into two plane-polarized components, one with the electric vector in the plane of incidence (“p-polarized”), and one with the electric vector normal to the plane of incidence (“s-polarized”). The refractive index of the incident light is therefore modified as follows:n=no/cos θo(p-polarization), and  (equation 10)n=no cos θo (s-polarization).  (equation 11)The transmittance and reflectance of the next interface, T2 and R2, are related to the real and imaginary components of E and H (i.e., E1, E2 and H1, H2) determined from equation 7:T2=(4nsn)/((E1+H1)2+(E2+H2)2)  (equation 12)R2=((E1−H1)2+(E2−H2)2/((E1+H1)2+(E2+H2)2)  (equation 13)
The total transmittance of the film and substrate at a given wavelength channel can be determined by summing the infinite series of the combined reflectance and transmittance terms, i.e., T1T2, T1T2R1R2, T1T2(R1R2)2, T1T2(R1R2)3, etc. The method for calculating total reflectance is performed in a like manner. This results in the following expressions for total transmittance T and reflectance R:T(λ, x, n1, ns, θo)=100(T1T2)/(1−R1R2),  (equation 14a)R(λ, x, n1, ns, θo)=100(R1′+T1′T1R2/(1−R1R2),  (equation 14b)where n1 and ns are the complex refractive indices of layer 1 and substrate s.
Repeat use of reference characters in the present specification and drawings is intended to represent same or analogous features or elements of the invention.